成组BFGS修正的紧凑形式与成组记忆修正算法

１．引言实际问题中经常要遇到一族函数极小值问题的求解，即ｍｉｎｆｉ（ｘ），ｉ＝１，...，Ｐ；（１．１）其中人：Ｒ"、Ｒ具有公共的Ｈｅｓｓｉａｎ矩阵Ｇ（ｘ）。７'ｆｉ（ｘ），ｒ是适中的数值．如在各种负载下的弹性体研究中，即要遇到问题（ｌ．Ｉ）的求解，其中人（Ｃ）一人Ｃ）＋ｑＣ十Ｃ；（＝１，...，...．对于不同的比则人（Ｘ）具有不同的极小点和不同的梯度Ｄ人（Ｘ），但具有相同的Ｈｅｓｓｉａｎ矩阵Ｇ（Ｘ）．１９９４年，Ｏ'Ｌｅａｒｙ等【'］把拟一Ｎｅｗｔｏｎ算法推广至成组形式（ｍｕｌｔｉＰｌｅｖｅｒｓｉｏ...，…

REPRESENTATION OF MULTIPLE BFGS'S UPDATINGMATRIX AND MULTIPLE VERSION OF BFGS'S METHODWITH LIMITED STORAGE

In 1994, O'leary and Yeremin extended the quasi-Newton method for minimizing a collection of functions with a common Hessian matrix to the block version,and discussed some algebraic properties of this block quasi-Newton method. In thispaper, we derive compact representations of the block BFGS's updating matrices.These representations allow us to efficiently implement limited memory methods,e.g., the limited memory BFGS method, for minimizing a collection of functionswith a common Hessian matrix. The method relieves the requirement for the storage counts and has the savings in the operation counts, in particular, for large scaleproblems. The numerical experiments for the multiple unconstrained optimizationproblems show that the method works efficiently. Compared with O'Leary's multiple version of BFGS method, our multiple version of the limited memory BFGSmethod is more efficient in the total operation counts and the storage counts.